Specific heat log-periodicity from multifractal energy spectra

Transcription

1 Physics Letters A 318 (2003) Specific heat log-periodicity from multifractal energy spectra Danyel J.B. Soares a, Marcelo L. Lyra b,, Luciano R. da Silva a a Departamento de Física, Universidade Federal do Rio Grande do Norte, Natal, RN, Brazil b Departamento de Física, Universidade Federal de Alagoas, Maceió, AL, Brazil Received 7 July 2003; accepted 29 August 2003 Communicated by J. Flouquet Abstract In this work, we investigate the emergence of log-periodic oscillations in the low-temperature behavior of the specific heat of systems whose energy spectra present a self-similar character. The critical attractor of z-generalized logistic maps are used to generate multifractal energy spectra with tunable singularity spectra. We study the relationship between the average value and amplitude of the log-periodic oscillations on the map nonlinearity strength as well as on the scaling exponents characterizing the energy spectrum. Our numerical results show a monotonic decrease of the oscillations amplitude with increasing nonlinearity. Further, we obtain that the average low-temperature specific heat is directly related to the minimum singularity strength governing the scaling behavior of the most concentrated energy range Elsevier B.V. All rights reserved. PACS: y; Hv; g Keywords: Multifractal energy spectral; Nonlinear maps; Log-periodic specific heat 1. Introduction Nonlinear one-dimensional dissipative maps have been widely used to describe some relevant aspects related to the emergence of complex behavior in nature. Usually the dynamical attractor presents a transition between periodic and chaotic behavior at which the map displays a strong exponential sensitivity to initial conditions [1]. At the onset of chaos, the attractor exhibits a multifractal structure with long-range temporal and spatial correlations [2]. Due to their simplicity, most of the scaling behavior characterizing the transi- * Corresponding author. address: marcelo@fis.ufal.br (M.L. Lyra). tion to chaos can be obtained with high accuracy and provide important insights on the behavior of more complex systems. The self-similar character of the critical attractor of low-dimensional dissipative maps has been explored to study some thermodynamic features of quasi-crystals. In general, quasi-crystals have properties that are intermediate between periodic and random structures [3,4]. In particular, the energy spectrum has a quite complex structure. Simplified fractals based in the Cantor set and Fibonacci sequence [5 8], as well as the critical attractor of the logistic and circle maps [9,10], have been used to model the energy spectrum of quasi-periodic systems. The thermodynamic behavior derived from such self-similar spectra display some anomalous features with the most prominent one being /$ see front matter 2003 Elsevier B.V. All rights reserved. doi: /j.physleta

2 D.J.B. Soares et al. / Physics Letters A 318 (2003) related to the emergence of log-periodic oscillations in the low-temperature behavior of the specific heat. A series of recent works have shown that the average low-temperature specific heat is intimately connected with some underlying fractal dimension characterizing the energy spectrum [6 8,10]. In the present work, we will contribute to this subject revealing additional features concerning the relationship between the low-temperature thermodynamic properties and the multifractal character of the energy spectra and the underlying nonlinearity. Specifically, we will consider the energy spectra and density of states generated by the family of z-generalized logistic maps. Within this family, the topological structure based on bifurcations is z-independent. However, the metrics defining all scaling exponents is z-dependent, thus allowing us to closely investigate the parametric relation between the average value and amplitude of the log-periodic oscillations of the specific heat and the scaling exponents characterizing the energy spectrum. where x n is a particular point of the critical attractor and min{x i } and max{x i } are the minimum and maximum values achieved by x n. Typically we generated 2 18 points in the attractor. In Fig. 1 we show the density of states (DOS) as obtained from the usual logistic map at a c (z = 2) = together with the integrated density of states (IDOS) for z = 1.1and z = 3.0. The self-similar character of the DOS is ex- 2. Energy spectra based on critical z-logistic maps In the following, we will consider a family of generalized logistic maps in the form x t+1 = 1 a x t z, (1) whose attractor displays a cascade of bifurcations converging to a critical point a c (z) after which chaotic orbits exist. The critical values a c (z) are known with high precision and can be found, for example, in [11]. At the critical point, the dynamical attractor can be characterized by a multifractal measure [2] with scaling exponents depending on the map inflexion z. The multifractality is fully characterized by the singularity spectrum f(α) which provides the fractal dimensions f of sets with singularity strength α. Usually, the singularity spectrum has a parabolic-like shape with minimum and maximum singularity strengths that correspond, respectively, to the exponents governing the scaling behavior in the most concentrated and rarefied regions of the attractor. Taking as a basic set the critical attractor of a z-generalized logistic map, we associate a normalized energy spectrum defined in the interval [0, 1] given by E n = x n min{x i } max{x i } min{x i }, (2) Fig. 1. (a) The density of states (DOS) derived from the usual logistic map (z = 2) at the onset of chaos. The energy of allowed states are defined in the text. The inset corresponds to an amplification of a small energy range which explicitly shows the scale invariant nature of the energy spectrum. (b) The integrated density of states (IDOS) derived from the generalized logistic maps with inflexions z = 1.1 and 3.0. The devil staircase shape reflects the presence of gaps on all energy scales. The slope in logarithmic scale correspond to the singularity strength α min of the most concentrated set of states.

3 454 D.J.B. Soares et al. / Physics Letters A 318 (2003) posed in the inset showing an amplification of a small energy range. The IDOS presents an average powerlaw behavior whose exponent corresponds to the minimum singularity strength of the critical attractor. The devil staircase shape is also a consequence of the scale invariant spectrum. These trends are common to maps with distinct inflexions although the power-law exponent of the IDOS is z-dependent. 3. The specific heat With the energy spectra generated using the procedure described in the last section, we compute the specific heat, using Boltzmann statistics ( ) [ 1 n C(T ) = E2 n e E n/t T 2 n e E n/t ( n E ne E ) ] n/t 2, (3) n e E n/t where we considered units of k B = 1, and thus dimensionless energy and temperature scales. In Fig. 2, we show the specific heat C(T ) obtained from the multifractal energy spectra derived from the z = 1.1 and z = 3.0 generalized logistic maps at the onset of Fig. 2. The specific heat C(T) of systems presenting multifractal energy spectra derived from the generalized logistic map with inflexions z = 1.1 (solid line) and 3.0 (dotted line). The log-periodic oscillations in the low-temperature regime reflects the scale invariance of the energy spectrum. The average value coincides with the minimum singularity strength α min which dominates the scaling behavior at the bottom of the energy band. The amplitude of these log-periodic oscillations decreases with increasing nonlinearity. However the period, in logarithmic temperature scale is roughly z-independent. chaos. In the high-temperature regime, C(T ) presents the usual 1/T 2 decay of bounded energy spectra for any inflexion z. At low-temperatures, the specific heat presents a sequence of Schottky peaks distributed in a log-periodic fashion in temperature scale. This feature is associated with the discrete scale invariant nature of the energy spectrum and its main characteristics have been explored in model systems with energy spectra derived from Cantor sets, Fibonacci sequences and iterated logistic and circle maps at the onset of chaos [5 10]. It has been well established that for monofractal energy spectra, as, for example, the one derived from single-scale Cantor sets, the specific heat oscillations are around the fractal dimension of the energy spectra. For energy spectra derived from multiscale Cantor sets, it has been demonstrated, both through scale arguments and numerically, that the oscillations are around one of the limiting dimensions of the multifractal singularity spectrum [8]. However, the factors controlling the amplitude of log-periodic oscillations are still not fully understood. For the present multifractal energy spectra derived from the family of z-logistic maps, we estimated the average value of the specific heat C at lowtemperatures for a wide range of z values. Our results are summarized in Table 1. Notice that C displays a nonmonotonic behavior as z is increased. We further noticed that, within our numerical accuracy, all estimated values for C coincide with the minimal singularity strength α min of the energy spectrum. This scaling exponent governs the scaling behavior of the most concentrated regions of the spectrum. The fact that C =α min means that the bottom of the energy band corresponds to one of the most concentrated sets of the energy spectrum. We also estimated the relative amplitude A = (C max C min )/ C of the log-periodic specific heat oscillations for distinct values of the logistic map inflexion z (see Table 1 and Fig. 3). These amplitudes monotonically decrease as z increases and, therefore, cannot be trivially correlated with α min. We explored its parametric dependence on the map nonlinearity z. We found that the relative amplitude reaches a maximum value of A max = 2 in the limit of very weak nonlinearity z 1. In this regime the log-periodic oscillations drive the specific heat to a minimum value very close to C min (T ) = 0 at some characteristic temperatures, which implies that the spectrum presents a

4 D.J.B. Soares et al. / Physics Letters A 318 (2003) Table 1 Critical parameters and specific heat data derived form z-generalized logistic maps. Values for the critical parameter a c, singularity strengths α min and α max and support fractal dimension d f are from Ref. [11]. Last two columns refer to the present estimates for the average value C(T ) of the specific heat in the low-temperature domain of log-periodic oscillations and the relative oscillation amplitudes A. Within our numerical accuracy, C(T ) coincides with the minimum singularity strength α min z a c α min α max d f C(T ) A = (C max C min )/ C(T) Fig. 3. The relative amplitude of the specific heat log-periodic oscillations as a function of the map inflexion. It achieves a maximum A max = 2 in the weakly non-linear limit z 1. The inset shows the asymptotic power-law decay of the oscillations amplitude. In the strongly non-linear regime z, our numerical results suggest that the relative amplitude scales as A 1/z 1/6. significant shortage of energy scales of the order of k B T. The amplitude of the log-periodic oscillations decreases with increasing nonlinearity. In the strongly nonlinear regime the asymptotic decay law can be fairly fitted by A 1/(z 1) 1/6. We have also observed that the period in logarithmic temperature scale of the log-periodic oscillations is roughly independent on the map inflexion z. of the specific heat of systems presenting a multifractal energy spectra. In the present study, the energy spectra were derived from the critical attractor of a family of generalized logistic maps. These spectra are more concentrated exactly at the first spectral branch and therefore, the low-temperature behavior of the thermodynamic properties are influenced by the fractal exponent α min governing the scaling behavior of the most concentrated set of states. We showed that the average value, around it the specific heat oscillates at low-temperatures, coincides with α min for any map inflexion z. However, the amplitude of these logperiodic oscillations are not directly correlated to the scaling behavior of a particular set of the energy spectrum. The log-periodic amplitudes are maximal in the weakly nonlinear regime and may drive the specific heat very near to vanishing values. On the other hand, these amplitudes vanishes in the asymptotic limit of z following a power-law decay. The period of these oscillations, in logarithmic temperature scale, is roughly independent of the map inflexion and therefore does not seem to be influenced by any scaling exponent. The above results remain valid for the family of periodic maps which belong to the same universality class of the z-logistic maps [11]. Acknowledgements 4. Summary and conclusions In summary, we studied the phenomenon of logperiodic oscillations in the temperature dependence The authors acknowledge the partial financial support of CNPq and CAPES (Brazilian research agencies) and FAPEAL (Alagoas State agency).

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